A note on the place topology of projective planes
type: Item
instance of: scholarly article
title: A note on the place topology of projective planes
zbMATH Open document ID: 0969.51020
DOI: 10.1023/A:1005288108409
author: Franz B. Kalhoff
published in: Geometriae Dedicata
publication date: 2001-10-07T00:00:00Z
review text: Every non-injective epimorphism \(\varphi: \Pi\to \Pi'\) between projective planes \(\Pi\), \(\Pi'\) induces a ``place topology'' \(\tau_\varphi\) on the point of set and on the line set of \(\Pi\) (by requiring that points and lines of \(\Pi'\) have open preimages, and that central projections in \(\Pi\) are continuous). The author addresses the question whether \((\Pi, \tau_\varphi)\) is a topological projective plane (in the sense of \textit{H. Salzmann} et al. [Compact projective planes, de Gruyter Berlin (1996; Zbl 0851.51003); this is know to be true if \(\Pi\) is a Moufang plane, or a kind of generalized Moulton plane [see \textit{P. Hartmann}, Geom. Dedicata 26, No. 3, 259-272 (1988; Zbl 0659.51020)] or if \(\varphi\) is induced by a uniform valuation [see \textit{F. B. Kalhoff}, Geom. Dedicata 54, No. 2, 199-224 (1995; Zbl 0839.51011)]. The author proves that \((\Pi,\tau_\varphi)\) is a non-discrete (totally disconnected) topological projective plane if \(\varphi\) is induced by a place \(\lambda: K\to K'\cup \{\infty\}\) between ternary fields \(K,K'\) such that the extended radical \(R_a(K)\) is bounded with respect to \(\tau_\varphi\), i.e. \(R_a (K) \cdot\lambda^{-1} (K')\) is a proper subset of \(K\) (the extended radical \(R_a (K)\) is a measure for the algebraic quality of \(K)\). The proof is achieved by coarsening \(\varphi\) to a ``friendly epimorphism'' in the sense von \textit{F. B. Kalhoff} [On epimorphisms and projectivities of projective planes, J. Geom., to appear] and by applying results of \textit{F. B. Kalhoff} [loc. cit.].
reviewed by: Theo Grundhöfer
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 51A35
zbMATH DE Number: 1546676
zbMATH Keywords: topological ternary field
zbMATH Keywords: place topology
zbMATH Keywords: topological projective plane
MaRDI profile type: Publication
full work available at URL: a:1005288108409
OpenAlex ID: W91273913
description: scientific article; zbMATH DE number 1546676